Inner-Product-Spaces.md

[Inner Product Spaces]Definition: Let (V,F)(V,F)(V,F) be a vector space. An inner product is a map of the form ⟨⋅,⋅⟩:V×V→F\langle\cdot,\cdot\rangle : V \times V \rightarrow F⟨⋅,⋅⟩:V×V→F such that for all x,y,z∈Vx,y,z \in Vx,y,z∈V and c∈Fc \in Fc∈F the following axioms hold:
(P1) ⟨x,y⟩=⟨y,x⟩‾\langle x,y \rangle = \overline{\langle y,x \rangle}⟨x,y⟩=⟨y,x⟩​ (conjugate symmetry)
(P2) a. ⟨x+y,z⟩=⟨x,z⟩+⟨y,z⟩\langle x+y,z \rangle = \langle x,z \rangle + \langle y,z \rangle⟨x+y,z⟩=⟨x,z⟩+⟨y,z⟩ (linearity in the first argument)
b. ⟨cx,y⟩=c⟨x,y⟩\langle cx,y \rangle = c\langle x,y \rangle⟨cx,y⟩=c⟨x,y⟩ (homogeneity in the first argument)
(P3) ⟨x,x⟩≥0\langle x,x \rangle \geq 0⟨x,x⟩≥0 and ⟨x,x⟩=0\langle x,x\rangle = 0⟨x,x⟩=0 iff x=0vx=0_vx=0v​ (positive definiteness)
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Theorem: "Cauchy-Schwarz Inequality"\textit{"Cauchy-Schwarz Inequality"}"Cauchy-Schwarz Inequality" Let (V,F)(V,F)(V,F) be a vector space with an inner product ⟨⋅,⋅⟩\langle\cdot,\cdot\rangle⟨⋅,⋅⟩. Then for all x,y∈Vx,y \in Vx,y∈V the following inequality holds:
∣⟨x,y⟩∣2≤⟨x,x⟩⟨y,y⟩|\langle x,y \rangle|^2 \leq \langle x,x \rangle \langle y,y \rangle∣⟨x,y⟩∣2≤⟨x,x⟩⟨y,y⟩Proof: Let x,y∈Vx,y \in Vx,y∈V and c∈Fc \in Fc∈F.

0≤⟨x−cy,x−cy⟩=⟨x,x⟩−⟨x,cy⟩−⟨cy,x⟩+⟨cy,cy⟩=⟨x,x⟩−c‾⟨x,y⟩−c⟨y,x⟩+∣c∣2⟨y,y⟩=⟨x,x⟩−c‾⟨x,y⟩−c⟨x,y⟩‾+∣c∣2⟨y,y⟩=⟨x,x⟩−2Re(c⟨x,y⟩)+∣c∣2⟨y,y⟩\large\begin{align*} 0 & \leq \langle x-cy,x-cy \rangle \\
& = \langle x,x \rangle - \langle x,cy \rangle - \langle cy,x \rangle + \langle cy,cy \rangle \\
& = \langle x,x \rangle - \overline{c}\langle x,y \rangle - c\langle y,x \rangle + |c|^2\langle y,y \rangle \\
& = \langle x,x \rangle - \overline{c}\langle x,y \rangle - c\overline{\langle x,y \rangle} + |c|^2\langle y,y \rangle \\
& = \langle x,x \rangle - 2Re(c\langle x,y \rangle) + |c|^2\langle y,y \rangle \\
\end{align*}0​≤⟨x−cy,x−cy⟩=⟨x,x⟩−⟨x,cy⟩−⟨cy,x⟩+⟨cy,cy⟩=⟨x,x⟩−c⟨x,y⟩−c⟨y,x⟩+∣c∣2⟨y,y⟩=⟨x,x⟩−c⟨x,y⟩−c⟨x,y⟩​+∣c∣2⟨y,y⟩=⟨x,x⟩−2Re(c⟨x,y⟩)+∣c∣2⟨y,y⟩​
#EE501 - Linear Systems Theory at METU